It's still the same meaning. But one is continuing with respect to time and the other is continuing with respect to location. In the first sentence the fight is continuing from an earlier time into afternoon
. In the second sentence the fight was being carried on in one location and moved into the parking lot. For example, perhaps a husband and wife start arguing with each other inside a store. They keep bickering through the check out line. And as they're moving to the car they keep on arguing. Then you could say,
The argument continued into the parking lot.
Perhaps a fight fight started between two individuals or even two groups. One group tries to move away from the other, but the other pursues. Then the fight continues from one location into another.